I am a bit bewildered when I try to think about empty space. Does it make sense to think about space insofar as it is space? What sort of existence, if any, does it have? Is it nothing?
Thank you!

There are two major views about space, and they give different answers to your question.

One view is "substantivalism." On this view, space really is a thing of a certain sort—a substance. Space would exist even if nothing else did. Needless to say, space it not like things as we usually think of them, but it has its own sort of reality. For Newton space was, among other things, a system of absolute positions. Newton believed that there was an absolute distinction between rest and motion, and that called for a corresponding system of positions. However, the points of space were otherwise indistinguishable; one point was intrinsically like any other.

In contemporary physics space and time are deeply intertwined, and we talk about space-time. Space time in general relativity is mathematically like a field (think of the electromagnetic field), and unlike Newtonian space, the points of space-time aren't all alike. This goes with the idea that space-time itself is curved. Roughly, the curvature at two different points of space-time (represented by a mathematical object called a tensor) can be different.

The alternative to the substantival view of space (or space-time) is relationalism. According to relationalists, space(-time) doesn't have any independent existence. To talk about space is really to talk about relationships among physical objects. For example: the fact that one object "takes up more space" than another" might simply amount to the fact that the second object would fit inside the first. The fact that A is further from B than from C might mean that if we had a large collection of rods whose ends would meet up if we brought them together (intuitively, are of the same length), we would have to lay more of those rods end-to-end to reach from A to B than to reach from A to C. More complicated variations on this idea lead to a relational account of the space-time of general relativity. On the relational view, there is no such thing as space or space-time, but there are spatial/geometric facts about the physical things that make up the world.